边界区域能量算子与散装运算符重构：模块化流程揭示

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本文主要探讨了模块化流程在量子信息领域中的重要应用，特别是针对纠缠楔（Entanglement Wedge）中的散装（Bulk）算子的重构问题。作者Thomas Faulkner和Aitor Lewkowycz是来自美国伊利诺伊大学厄巴纳-香槟分校和斯坦福大学理论物理研究所的学者，他们结合边界区域的低能算子（Flow Energy Operators）来研究这一现象。在散装与边界区域的联系中，关键的概念是模块化流程（Modular Flow）。在量子场论中，模块化演化是一种描述量子系统中信息传播和量子态结构的重要工具。作者指出，通过按照N（可能指的是系统的大小或者维度）的主导顺序，他们能够在边界子区域内的单迹线（Single Trace）运算符上构建出双边界（Double Boundary）运算符，这反映了模块化的局部性原理。文章的核心发现是，散装算子的重建可以通过边界算子的模块化流程来实现。这意味着在理解散装模块化流程的作用的基础上，可以找到散装算子的精确公式，尽管这个任务在实践中可能会面临一些挑战，但理论上是可行的。这种等价性不仅揭示了理论模型中的深层次关系，也为我们提供了研究复杂量子系统的新视角。在特定情况下，当散装算子位于 Ryu-Takayanagi 表面时，作者给出了一个表达式，这个表达式仅依赖于边界信息，进一步展示了模块化流程在描述量子纠缠和信息传递中的关键角色。这种结果对于理解量子引力中的AdS/CFT对偶（Anti-de Sitter/Conformal Field Theory correspondence）以及在高维量子场论和量子信息科学中的其他理论应用具有重要意义。这篇论文不仅深化了我们对量子系统的模块化结构的理解，也为处理散装和边界之间的复杂相互作用提供了一种计算工具，对于推动量子信息科学和量子引力理论的发展具有重要的学术价值。

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JHEP07(2017)151

the modes. In general, G

(x, y) is not translation invariant and one can’t simplify the

modes further.

As a side note, the generator of the full modular ﬂow U(t) = e

−i(K

−K

is well deﬁned

in the continuum limit: it acts like the usual modular ﬂow when acting in an operator in R

R but it is also an good operator, which doesn’t depend on details near the entangling

surface ∂R. In this way, sometimes is useful to think of the previous correlators in terms of:

hAU(t)Bi (2.6)

which is equivalent to the previous discussion when A, B are in the same subregion, but it

is well deﬁned more generally.

2.1 Gaussian states in free ﬁeld theory

Gaussian states in free theories have been explored extensively in the literature (see for ex-

ample [17] and references therein). Since all correlators are ﬁxed by the two point function,

the density matrix will be gaussian and the modular hamiltonian bilinear. We will be focus-

ing on scalar ﬁelds, the operators in the algebra of the region R,

are {Φ(X), Π(Y ); ∀x ∈ R}

and the respective modular hamiltonian will be:

Φ(X).K(X, Y ).

Φ(Y ) (2.7)

Φ(X) ≡ (Φ(X), Π(X)) , Π(X) = n

∂

Φ(X)

where n

is the unit term to the Cauchy slice containing R.

In general, K is unknown unless the system is specially symmetric: one can write it

formally in terms of matrix inversions and log’s of correlators, but these are hard to obtain

in the continuum limit (and only understandable in a distributional sense) although one

can obtain it numerically using a lattice. Free ﬁelds are interesting in the holographic

setting because to leading order in 1/N, the bulk quantum theory consists of free ﬁelds.

Subleading orders introduce weak interactions that can be included perturbatively in the

previous discussion: for example if we have a Φ

interaction, there will be a contribution

to the modular hamiltonian which will be trilinear on the ﬁelds.

The modular fourier modes will then be a linear combination of Φ, Π:

(X) =

dY (c

(X, Y )Π(Y ) − n

∂

(X, Y )Φ(Y )) (2.8)

where the coeﬃcients are determined by the canonical commutation relations:

(X, Y ) = −i(e

2πw

− 1)hΦ

(X)Φ(Y )i.

These modular frequency modes can also be used as a basis of operators. This is how

the entanglement entropy is computed in [17]. The idea is that instead of labeling the

modes by (d −1) coordinates x, we can label them by w and (d −2) coordinates X

, which

correspond to a codimension 2 surface S ⊂ R. If we think of the c

, Y ) as a matrix

This is a ﬁeld theoretical discussion, but in the next section, when we apply this to holography, this

will be a subregion of the bulk.

– 5 –

JHEP07(2017)151

∝ G

, Y ) with index Y

= (w, X

), then diﬀerent Y, Y

have diﬀerent values and

is always diﬀerent from zero (we don’t expect the correlators to vanish). Furthermore,

no linear combination of Φ,Π can give zero, so this matrix is invertible. The invertibility of

this matrix guarantees that Φ

) is a good basis, ie there is a Bogoliubov transformation

that sends {Φ(Y ), Π(Y )} → {Φ

), Φ

†

)}. This basis is useful because the modular

hamiltonian is simpler and the entropy can be computed easily.

In terms of the modes Φ

), the modular hamiltonian will read:

w>0

, Y

)Φ

−w

) (2.9)

This hamiltonian is slightly simpler compared with (2.7): it has one less integral. The

commutation relations determine K in terms of G

, Φ

)] = wΦ

) = w

, Y

, Z

)Φ

)

→

, Y

, Z

) = δ(X

− Z

) (2.10)

where we should think of K, the modular hamiltonian kernel as a distribution, which can be

thought as the matrix inverse of G

. Of course, this discussion seems very formal because

in order to deﬁne the modular frequency modes, one has to know the modular hamiltonian

in the ﬁrst place. However, as a pratical tool, it seems like it might be simpler to compute

(X, Y ) than evaluating the modular hamiltonian (2.7) in terms of the Φ, Π directly. The

reason is that this two point function has a very particular analytic structure: it picks up

a phase when going around the entangling surface ∂R. Solving the wave equation with

these branch cut boundary condition seems like an easier way to handle the problem and

this is how [18] dealt with the entanglement entropy massive scalar ﬁeld and a spherical

region. This approach to the entanglement entropy of free ﬁelds seems to be practically

useful also for less generic regions [19].

3 A smearing function in the entanglement wedge

The extrapolate dictionary relates boundary and bulk operators in the boundary of AdS:

lim

z→0

∆

Φ(x, z) = O(x) (3.1)

In the future, we will denote this as Φ(x, z = 0) = O(x). Given that the bulk ﬁeld is a free

ﬁeld, one can use the equations of motion to move this ﬁeld to the bulk, at the expense of

including more boundary operators:

Φ(X) =

dxf(X|x)O(x) (3.2)

One normally presents f(X|y) as the Green’s function for the scalar ﬁeld with support

at spacelike separation. However, f(X|x) can be understood by formally inverting the

– 6 –

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